Prime Numbers and Shizits
|
|
|
- Myrtle Watkins
- 5 years ago
- Views:
Transcription
1 Prime Numbers and Shizits Nick Handrick, Chris Magyar University of Wisconsin Eau Claire March 7, 2017
2 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers.
3 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C.
4 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE.
5 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.
6 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not..
7 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.. Let EF not be prime; therefore it is measured by some prime number G.
8 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.. Let EF not be prime; therefore it is measured by some prime number G..
9 Book IX: Proposition 20 Euclid, cerca 300 BCE Prime numbers are more than any assigned multitude of prime numbers. Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF = DE + DF is either prime or not.. Let EF not be prime; therefore it is measured by some prime number G.. Therefore, the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C.
10 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes?
11 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1)
12 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2
13 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3
14 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3.
15 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3. Once you reach N, every remaining number in your set is prime as well!
16 Prime sieve (the sieve of Eratosthenes) How does one efficiently find primes? Start with a set of natural numbers up to N (except 1) Mark the smallest number (2), and remove all multiples of 2 Mark the smallest number (3), and remove all multiples of 3. Once you reach N, every remaining number in your set is prime as well!
17 Prime Distribution / Probability range primes probability [1, 30]
18 Prime Distribution / Probability range primes probability 1 [1, 30] [1, 100] 25 4
19 Prime Distribution / Probability range primes probability 1 [1, 30] [1, 100] 25 4 [1, 1000]
20 Prime Distribution / Probability range primes probability 1 [1, 30] [1, 100] 25 4 [1, 1000] [1, ]
21 Prime Distribution / Probability range primes probability 1 [1, 30] 10 3 [1, 100] [1, 1000] [1, ] [1, ]
22 Primes and Goldbach An interesting graph (not entirely related but still awesome) Number of ways even numbers can be expressed as the sum of two primes:
23 Primes in History Carl Friedrich Gauss ( )
24 Primes in History Carl Friedrich Gauss ( ) G(x) X number of digits of X
25 x/ log x Yet another approximation for π(x) Gauss investigated π(x), and used some intuition to produce an approximation The intuition he had was that the odds some number X is prime is inversely proportional to the number of digits of X. π(x) X the number of digits of X X log X
26 Log Integral Function What is Li(x)? The Logarithmic Integral, also known as Li(x), is another mysterious way to approximate π(x) Definition: Li(x) = We ll see a graph of it soon x 2 1 log t dt
27 Here s that graph I promised
28 Here s that graph I promised The red staircase is π(x)
29 Here s that graph I promised The red staircase is π(x) The curve above it is Li(x)
30 Here s that graph I promised The red staircase is π(x) The curve above it is Li(x) The curve below it is x log x
31 Here s that graph I promised The red staircase is π(x) The curve above it is Li(x) The curve below it is x log x The graphs are a Good Fit approximation of each other
32 Good Fit approximation What does it even mean, rigorously?
33 Good Fit approximation What does it even mean, rigorously? They approach each other of course!
34 Good Fit approximation What does it even mean, rigorously? They approach each other of course! The log integral function converges much faster, but is more computationally intensive The Prime Number Theorem says that Li(x) and π(x) go to inifinity at the same rate
35 Riemann Hypothesis For any real number X, the number of prime numbers less than X is approximately Li(X).
36 Riemann Hypothesis For any real number X, the number of prime numbers less than X is approximately Li(X). Li(X) π(x) X log(x) for all X 2.01
37 Square Root Error Random error and random walks According to Barry Mazur, Statisticians use square root error as a gold standard for empirical data accuracy
38 ψ(x) and Riemann Hypothesis Second Formulation We talked earlier about ψ(x), but what does it have to do with the RH? ψ(x) = log p p k x = x ρ x ρ ρ log(2π) x where ρ are ALL zeroes of the zeta function You prove: ζ (s) ζ(s) = log(2π)
The Riemann Hypothesis
University of Hawai i at Mānoa January 26, 2016 The distribution of primes Euclid In ancient Greek times, Euclid s Elements already answered the question: Q: How many primes are there? Euclid In ancient
God may not play dice with the universe, but something strange is going on with the prime numbers.
Primes: Definitions God may not play dice with the universe, but something strange is going on with the prime numbers. P. Erdös (attributed by Carl Pomerance) Def: A prime integer is a number whose only
PRIMES. Barry Mazur. April 26, (A discussion of Primes: What is Riemann s Hypothesis?, the book I m currently writing with William Stein)
PRIMES Barry Mazur April 26, 2014 (A discussion of Primes: What is Riemann s Hypothesis?, the book I m currently writing with William Stein) William: https://vimeo.com/90380011 Figure: William The impact
A Simple Counterexample to Havil s Reformulation of the Riemann Hypothesis
A Simple Counterexample to Havil s Reformulation of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 0025 jsondow@alumni.princeton.edu The Riemann Hypothesis (RH) is the greatest
The Riemann Hypothesis
The Riemann Hypothesis by Pedro Pablo Pérez Velasco, 2011 Introduction Consider the series of questions: How many prime numbers (2, 3, 5, 7,...) are there less than 100, less than 10.000, less than 1.000.000?
Riemann s Zeta Function and the Prime Number Theorem
Riemann s Zeta Function and the Prime Number Theorem Dan Nichols nichols@math.umass.edu University of Massachusetts Dec. 7, 2016 Let s begin with the Basel problem, first posed in 1644 by Mengoli. Find
The Prime Number Theorem
The Prime Number Theorem We study the distribution of primes via the function π(x) = the number of primes x 6 5 4 3 2 2 3 4 5 6 7 8 9 0 2 3 4 5 2 It s easier to draw this way: π(x) = the number of primes
Turing and the Riemann zeta function
Turing and the Riemann zeta function Andrew Odlyzko School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ odlyzko May 11, 2012 Andrew Odlyzko ( School of Mathematics University
The Prime Number Theorem
Chapter 3 The Prime Number Theorem This chapter gives without proof the two basic results of analytic number theory. 3.1 The Theorem Recall that if f(x), g(x) are two real-valued functions, we write to
Here is another characterization of prime numbers.
Here is another characterization of prime numbers. Theorem p is prime it has no divisors d that satisfy < d p. Proof [ ] If p is prime then it has no divisors d that satisfy < d < p, so clearly no divisor
arxiv: v1 [math.gm] 11 Jun 2012
![arxiv: v1 [math.gm] 11 Jun 2012](/thumbs/90/104462535.jpg "arxiv: v1 [math.gm] 11 Jun 2012")
AM Comp. Sys. -9 Author version Dynamical Sieve of Eratosthenes Research arxiv:20.279v math.gm] Jun 202 Luis A. Mateos AM Computer Systems Research, 070 Vienna, Austria Abstract: In this document, prime
IMPROVING RIEMANN PRIME COUNTING
IMPROVING RIEMANN PRIME COUNTING MICHEL PLANAT AND PATRICK SOLÉ arxiv:1410.1083v1 [math.nt] 4 Oct 2014 Abstract. Prime number theorem asserts that (at large x) the prime counting function π(x) is approximately
The Prime Unsolved Problem in Mathematics
The Prime Unsolved Problem in Mathematics March, 0 Augustana Math Club 859: Bernard Riemann proposes the Riemann Hypothesis. 859: Bernard Riemann proposes the Riemann Hypothesis. 866: Riemann dies at age
Function estimating number of pairs of primes (p, q) for all z N of form z = p + q
Function estimating number of pairs of primes (p, q) for all z N of form z = p + q Dmitri Abramov ddmitri.abramov@gmail.com October 4, 203 Abstract This paper derives a function that estimates number of
Primes. Rational, Gaussian, Industrial Strength, etc. Robert Campbell 11/29/2010 1
Primes Rational, Gaussian, Industrial Strength, etc Robert Campbell 11/29/2010 1 Primes and Theory Number Theory to Abstract Algebra History Euclid to Wiles Computation pencil to supercomputer Practical
Twin primes (seem to be) more random than primes
Twin primes (seem to be) more random than primes Richard P. Brent Australian National University and University of Newcastle 25 October 2014 Primes and twin primes Abstract Cramér s probabilistic model
Riemann Hypotheses. Alex J. Best 4/2/2014. WMS Talks
Riemann Hypotheses Alex J. Best WMS Talks 4/2/2014 In this talk: 1 Introduction 2 The original hypothesis 3 Zeta functions for graphs 4 More assorted zetas 5 Back to number theory 6 Conclusion The Riemann
MATH3500 The 6th Millennium Prize Problem. The 6th Millennium Prize Problem
MATH3500 The 6th Millennium Prize Problem RH Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime
A Painless Overview of the Riemann Hypothesis
A Painless Overview of the Riemann Hypothesis [Proof Omitted] Peter Lynch School of Mathematics & Statistics University College Dublin Irish Mathematical Society 40 th Anniversary Meeting 20 December 2016.
On Legendre s formula and distribution of prime numbers
On Legendre s formula and distribution of prime numbers To Hiroshi, Akiko, and Yuko Kazuo AKIYAMA Tatsuhiko NAGASAWA 1 Abstract The conclusion in this paper is based on the several idea obtained in recherché
A Painless Overview of the Riemann Hypothesis
A Painless Overview of the Riemann Hypothesis [Proof Omitted] Peter Lynch School of Mathematics & Statistics University College Dublin Irish Mathematical Society 40 th Anniversary Meeting 20 December 2016.
Prime Numbers. Prime Numbers. Ramesh Sreekantan ISI, Bangalore. December 1, 2014
Prime Numbers Prime Numbers Ramesh Sreekantan ISI, Bangalore December 1, 2014 Introduction - Prime Numbers Prime numbers are the numbers 2,3,5,7... These are natural numbers which cannot be divided by
Why is the Riemann Hypothesis Important?
Why is the Riemann Hypothesis Important? Keith Conrad University of Connecticut August 11, 2016 1859: Riemann s Address to the Berlin Academy of Sciences The Zeta-function For s C with Re(s) > 1, set ζ(s)
The Evolution of the Circle Method in Additive Prime Number Theory
Lee University The Evolution of the Circle Method in Additive Prime Number Theory William Cole wcole00@leeu.edu 09 Aviara Dr. Advance, NC 27006 supervised by Dr. Laura Singletary April 4, 206 Introduction
Calculus II : Prime suspect
Calculus II : Prime suspect January 31, 2007 TEAM MEMBERS An integer p > 1 is prime if its only positive divisors are 1 and p. In his 300 BC masterpiece Elements Euclid proved that there are infinitely
Proposed Proof of Riemann Hypothesis
Proposed Proof of Riemann Hypothesis Aron Palmer Abstract Proposed proof of the Riemann hypothesis showing that positive decreasing continuous function which tends to zero as t goes to infinity can t have
Riemann Zeta Function and Prime Number Distribution
Riemann Zeta Function and Prime Number Distribution Mingrui Xu June 2, 24 Contents Introduction 2 2 Definition of zeta function and Functional Equation 2 2. Definition and Euler Product....................
PRIME NUMBERS YANKI LEKILI
PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume
Dramatis Personae. Robin Whitty. Riemann s Hypothesis, Rewley House, 15 September Queen Mary University of London
Queen Mary University of London Riemann s Hypothesis, Rewley House, 15 September 2013 Symbols Z = {..., 3, 2, 1,0,1,2,3,4,...} the integers R = the real numbers integers, fractions, irrationals C = the
Chapter 5: The Integers
c Dr Oksana Shatalov, Fall 2014 1 Chapter 5: The Integers 5.1: Axioms and Basic Properties Operations on the set of integers, Z: addition and multiplication with the following properties: A1. Addition
Newman s Conjecture for function field L-functions
Newman s Conjecture for function field L-functions David Mehrle Presenters: t < Λ t Λ Λ Joseph Stahl September 28, 2014 Joint work with: Tomer Reiter, Dylan Yott Advisors: Steve Miller, Alan Chang The
Moments of the Riemann Zeta Function and Random Matrix Theory. Chris Hughes
Moments of the Riemann Zeta Function and Random Matrix Theory Chris Hughes Overview We will use the characteristic polynomial of a random unitary matrix to model the Riemann zeta function. Using this,
5: The Integers (An introduction to Number Theory)
c Oksana Shatalov, Spring 2017 1 5: The Integers (An introduction to Number Theory) The Well Ordering Principle: Every nonempty subset on Z + has a smallest element; that is, if S is a nonempty subset
THE FORMATION OF PRIME NUMBERS THE SOLUTION FOR GOLDBACH S CONJECTURES
Research article THE FORMATION OF PRIME NUMBERS AND THE SOLUTION FOR GOLDBACH S CONJECTURES JOSÉ WILLIAM PORRAS-FERREIRA CARLOS ALBERTO ANDRADE Escuela Naval de Cadetes Almirante Padilla, Isla Manzanillo,
1 Euler s idea: revisiting the infinitude of primes
8.785: Analytic Number Theory, MIT, spring 27 (K.S. Kedlaya) The prime number theorem Most of my handouts will come with exercises attached; see the web site for the due dates. (For example, these are
The Riemann Hypothesis
The Riemann Hypothesis Matilde N. Laĺın GAME Seminar, Special Series, History of Mathematics University of Alberta mlalin@math.ualberta.ca http://www.math.ualberta.ca/~mlalin March 5, 2008 Matilde N. Laĺın
Disproof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Disproof Fall / 16
Disproof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Disproof Fall 2014 1 / 16 Outline 1 s 2 Disproving Universal Statements: Counterexamples 3 Disproving Existence
Notes on the Riemann Zeta Function
Notes on the Riemann Zeta Function March 9, 5 The Zeta Function. Definition and Analyticity The Riemann zeta function is defined for Re(s) > as follows: ζ(s) = n n s. The fact that this function is analytic
Elementary Number Theory and Cryptography, 2014
Elementary Number Theory and Cryptography, 2014 1 Basic Properties of the Integers Z and the rationals Q. Notation. By Z we denote the set of integer numbers and by Q we denote the set of rational numbers.
2 Elementary number theory
2 Elementary number theory 2.1 Introduction Elementary number theory is concerned with properties of the integers. Hence we shall be interested in the following sets: The set if integers {... 2, 1,0,1,2,3,...},
Riemann Hypothesis Elementary Discussion
Progress in Applied Mathematics Vol. 6, No., 03, pp. [74 80] DOI: 0.3968/j.pam.955803060.58 ISSN 95-5X [Print] ISSN 95-58 [Online] www.cscanada.net www.cscanada.org Riemann Hypothesis Elementary Discussion
CIMPA/ICTP research school. Prime number theory in F q [t]
![CIMPA/ICTP research school. Prime number theory in F q [t]](/thumbs/72/67395180.jpg "CIMPA/ICTP research school. Prime number theory in F q [t]")
CIMPA/ICTP research school Prime Prime F q [t] University of Georgia July 2013 1 / 85 Prime The story of the prime Prime Let π(x) = #{p x : p prime}. For example, π(2.9) = 1, π(3) = 2, and π(100) = 25.
Math Real Analysis
1 / 28 Math 370 - Real Analysis G.Pugh Sep 3 2013 Real Analysis 2 / 28 3 / 28 What is Real Analysis? Wikipedia: Real analysis... has its beginnings in the rigorous formulation of calculus. It is a branch
Proof of Lagarias s Elementary Version of the Riemann Hypothesis.
Proof of Lagarias s Elementary Version of the Riemann Hypothesis. Stephen Marshall 27 Nov 2018 Abstract The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative
How important is this hypothesis?
1 On the number of composite numbers less than a given value. Lemmas. Paper II: A new research perspective in the prime number theory Let us start with recalling the theorem presented in Part I: Below
An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p.
Chapter 6 Prime Numbers Part VI of PJE. Definition and Fundamental Results Definition. (PJE definition 23.1.1) An integer p is prime if p > 1 and p has exactly two positive divisors, 1 and p. If n > 1
Modelling large values of L-functions
Modelling large values of L-functions How big can things get? Christopher Hughes Exeter, 20 th January 2016 Christopher Hughes (University of York) Modelling large values of L-functions Exeter, 20 th January
Lecture 3. The Population Variance. The population variance, denoted σ 2, is the sum. of the squared deviations about the population
Lecture 5 1 Lecture 3 The Population Variance The population variance, denoted σ 2, is the sum of the squared deviations about the population mean divided by the number of observations in the population,
Prime Number Theory and the Riemann Zeta-Function
5262589 - Recent Perspectives in Random Matrix Theory and Number Theory Prime Number Theory and the Riemann Zeta-Function D.R. Heath-Brown Primes An integer p N is said to be prime if p and there is no
Primes, queues and random matrices
Primes, queues and random matrices Peter Forrester, M&S, University of Melbourne Outline Counting primes Counting Riemann zeros Random matrices and their predictive powers Queues 1 / 25 INI Programme RMA
February Fourier talks, Zeros of some self-reciprocal polynomials
February Fourier talks, 2011 Zeros of some self-reciprocal polynomials D. Joyner, USNA FFT 2011 at the Norbert Wiener Center, UMCP February 15, 2011 D. Joyner, USNA Zeros of some self-reciprocal polynomials
CHAPTER 3. Number Theory
CHAPTER 3 Number Theory 1. Factors or not According to Carl Friedrich Gauss (1777-1855) mathematics is the queen of sciences and number theory is the queen of mathematics, where queen stands for elevated
Number Theory and Algebraic Equations. Odile Marie-Thérèse Pons
Number Theory and Algebraic Equations Odile Marie-Thérèse Pons Published by Science Publishing Group 548 Fashion Avenue New York, NY 10018, U.S.A. http://www.sciencepublishinggroup.com ISBN: 978-1-940366-74-6
SOME FAMOUS UNSOLVED PROBLEMS. November 18, / 5
SOME FAMOUS UNSOLVED PROBLEMS November 18, 2014 1 / 5 SOME FAMOUS UNSOLVED PROBLEMS Goldbach conjecture: Every even number greater than four is the sum of two primes. November 18, 2014 1 / 5 SOME FAMOUS
Needles and Numbers. The Buffon Needle Experiment
eedles and umbers This excursion into analytic number theory is intended to complement the approach of our textbook, which emphasizes the algebraic theory of numbers. At some points, our presentation lacks
= (, ) V λ (1) λ λ ( + + ) P = [ ( ), (1)] ( ) ( ) = ( ) ( ) ( 0 ) ( 0 ) = ( 0 ) ( 0 ) 0 ( 0 ) ( ( 0 )) ( ( 0 )) = ( ( 0 )) ( ( 0 )) ( + ( 0 )) ( + ( 0 )) = ( + ( 0 )) ( ( 0 )) P V V V V V P V P V V V
MATH 25 CLASS 8 NOTES, OCT
MATH 25 CLASS 8 NOTES, OCT 7 20 Contents. Prime number races 2. Special kinds of prime numbers: Fermat and Mersenne numbers 2 3. Fermat numbers 3. Prime number races We proved that there were infinitely
The Origin of the Prime Number Theorem
Ursinus College Digital Commons @ Ursinus College Number Theory Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Fall 2018 The Origin of the Prime Number
Lecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method. Daniel Goldston
Lecture 1: Small Prime Gaps: From the Riemann Zeta-Function and Pair Correlation to the Circle Method Daniel Goldston π(x): The number of primes x. The prime number theorem: π(x) x log x, as x. The average
arxiv: v1 [math.nt] 31 Mar 2011
![arxiv: v1 [math.nt] 31 Mar 2011](/thumbs/71/66009929.jpg "arxiv: v1 [math.nt] 31 Mar 2011")
EXPERIMENTS WITH ZETA ZEROS AND PERRON S FORMULA ROBERT BAILLIE arxiv:03.66v [math.nt] 3 Mar 0 Abstract. Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to
THE RIEMANN HYPOTHESIS: IS TRUE!!
THE RIEMANN HYPOTHESIS: IS TRUE!! Ayman MACHHIDAN Morocco Ayman.mach@gmail.com Abstract : The Riemann s zeta function is defined by = for complex value s, this formula have sense only for >. This function
The Fundamental Theorem of Arithmetic
Chapter 1 The Fundamental Theorem of Arithmetic 1.1 Primes Definition 1.1. We say that p N is prime if it has just two factors in N, 1 and p itself. Number theory might be described as the study of the
Discrete Structures Lecture Primes and Greatest Common Divisor
DEFINITION 1 EXAMPLE 1.1 EXAMPLE 1.2 An integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite.
First, let me recall the formula I want to prove. Again, ψ is the function. ψ(x) = n<x
8.785: Analytic Number heory, MI, spring 007 (K.S. Kedlaya) von Mangoldt s formula In this unit, we derive von Mangoldt s formula estimating ψ(x) x in terms of the critical zeroes of the Riemann zeta function.
7. Prime Numbers Part VI of PJE
7. Prime Numbers Part VI of PJE 7.1 Definition (p.277) A positive integer n is prime when n > 1 and the only divisors are ±1 and +n. That is D (n) = { n 1 1 n}. Otherwise n > 1 is said to be composite.
The New Book of Prime Number Records
Paulo Ribenboim The New Book of Prime Number Records Springer Contents Preface Guiding the Reader Index of Notations ix xv xvii Introduction 1 CHARTER 1 How Many Prime Numbers Are There? 3 I. Euclid's
Computing Bernoulli Numbers Quickly
Computing Bernoulli Numbers Quickly Kevin J. McGown December 8, 2005 The Bernoulli numbers are defined via the coefficients of the power series expansion of t/(e t ). Namely, for integers m 0 we define
Lecture 1: Riemann, Dedekind, Selberg, and Ihara Zetas
Lecture 1: Riemann, Dedekind, Selberg, and Ihara Zetas Audrey Terras U.C.S.D. 2008 more details can be found in my webpage: www.math.ucsd.edu /~aterras/ newbook.pdf First the Riemann Zeta 1 The Riemann
Big doings with small g a p s
Big doings with small g a p s Paul Pollack University of Georgia AAAS Annual Meeting February 16, 2015 1 of 26 SMALL GAPS: A SHORT SURVEY 2 of 26 300 BCE Let p n denote the nth prime number, so p 1 = 2,
Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves
Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves William Stein University of California, San Diego http://modular.fas.harvard.edu/ Bremen: July 2005 1 This talk reports
What do we actually know about prime numbers distribution?
1 On the number of composite numbers less than a given value. Lemmas, continued. Paper III: What do we know about prime numbers distribution? Paper II presented 3 of 7 lemmas that confirm the conjecture
Unit 1. Number Theory
Unit 1 Number Theory 1-1 Divisibility Rules Divisible by: Rule 2 The number is even (it ends in 0, 2, 4, 6 or 8) 3 The sum of its digits is divisible by 3 (eg 741: 7 + 4 + 1 = 12) 4 The last two digits
x #{ p=prime p x }, as x logx
1 The Riemann zeta function for Re(s) > 1 ζ s -s ( ) -1 2 duality between primes & complex zeros of zeta using Hadamard product over zeros prime number theorem x #{ p=prime p x }, as x logx statistics
Prime Number Diffeomorphisms, Diophantine Equations and the Riemann Hypothesis
Prime Number Diffeomorphisms, Diophantine Equations and the Riemann Hypothesis arxiv:math-ph/0411071v1 Nov 004 Lubomir Alexandrov and Lachezar Georgiev Institute for Nuclear Research and Nuclear Energy
arxiv:math/ v4 [math.ho] 22 Sep 2003
![arxiv:math/ v4 [math.ho] 22 Sep 2003](/thumbs/94/120599779.jpg "arxiv:math/ v4 [math.ho] 22 Sep 2003")
Thoughts on the Riemann Hypothesis arxiv:math/0306042v4 [math.ho] 22 Sep 2003 G. J. Chaitin, IBM Research http://www.cs.auckland.ac.nz/cdmtcs/chaitin The simultaneous appearance in May 2003 of four books
Section X.55. Cyclotomic Extensions
X.55 Cyclotomic Extensions 1 Section X.55. Cyclotomic Extensions Note. In this section we return to a consideration of roots of unity and consider again the cyclic group of roots of unity as encountered
Congruent Number Problem and Elliptic curves
Congruent Number Problem and Elliptic curves December 12, 2010 Contents 1 Congruent Number problem 2 1.1 1 is not a congruent number.................................. 2 2 Certain Elliptic Curves 4 3 Using
Towards the Twin Prime Conjecture
A talk given at the NCTS (Hsinchu, Taiwan, August 6, 2014) and Northwest Univ. (Xi an, October 26, 2014) and Center for Combinatorics, Nankai Univ. (Tianjin, Nov. 3, 2014) Towards the Twin Prime Conjecture
INFORMATION-THEORETIC EQUIVALENT OF RIEMANN HYPOTHESIS
INFORMATION-THEORETIC EQUIVALENT OF RIEMANN HYPOTHESIS K. K. NAMBIAR ABSTRACT. Riemann Hypothesis is viewed as a statement about the capacity of a communication channel as defined by Shannon. 1. Introduction
Part II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
Elementary Number Theory
Elementary Number Theory 21.8.2013 Overview The course discusses properties of numbers, the most basic mathematical objects. We are going to follow the book: David Burton: Elementary Number Theory What
Intermediate Math Circles March 6, 2013 Number Theory I
What is Number Theory? Intermediate Math Circles March 6, 01 Number Theory I A branch of mathematics where mathematicians examine and study patterns found within the natural number set (positive integers).
Analytic. Number Theory. Exploring the Anatomy of Integers. Jean-Marie. De Koninck. Florian Luca. ffk li? Graduate Studies.
Analytic Number Theory Exploring the Anatomy of Integers Jean-Marie Florian Luca De Koninck Graduate Studies in Mathematics Volume 134 ffk li? American Mathematical Society Providence, Rhode Island Preface
EXPLICIT RESULTS ON PRIMES. ALLYSA LUMLEY Bachelor of Science, University of Lethbridge, 2010
EXPLICIT RESULTS ON PRIMES. ALLYSA LUMLEY Bachelor of Science, University of Lethbridge, 200 A Thesis Submitted to the School of Graduate Studies of the University of Lethbridge in Partial Fulfillment
On the Langlands Program
On the Langlands Program John Rognes Colloquium talk, May 4th 2018 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for
Euclid-Euler-Jiang Prime Theorem
Euclid-Euler-iang Prime Theorem Chun-Xuan iang POBox 394 Beiing 00854 P R China iangchunxuan@sohucom Abstract Santilli s prime chains: P + ap b L ( a b ab If λn a P λ LP n P LPn b we have ( as There exist
On Exponentially Perfect Numbers Relatively Prime to 15
On Exponentially Perfect Numbers Relatively Prime to 15 by Joseph F. Kolenick Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Mathematics Program YOUNGSTOWN
Formulae for Computing Logarithmic Integral Function ( )!
Formulae for Computing Logarithmic Integral Function x 2 ln t Li(x) dt Amrik Singh Nimbran 6, Polo Road, Patna, INDIA Email: simnimas@yahoo.co.in Abstract: The prime number theorem states that the number
Goldbach and twin prime conjectures implied by strong Goldbach number sequence
Goldbach and twin prime conjectures implied by strong Goldbach number sequence Pingyuan Zhou E-mail:zhoupingyuan49@hotmail.com Abstract In this note, we present a new and direct approach to prove the Goldbach
The Mertens conjecture revisited
The Mertens conjecture revisited Herman te Riele, CWI, Amsterdam (joint work with Tadej Kotnik, Univ. of Ljubljana, Slovenia) 7th Algorithmic Number Theory Symposium Technische Universität Berlin, 23 28
9/21/2018. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions. Properties of Functions
How can we prove that a function f is one-to-one? Whenever you want to prove something, first take a look at the relevant definition(s): x, y A (f(x) = f(y) x = y) f:r R f(x) = x 2 Disproof by counterexample:
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Spring 2006
Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 1 / 1 Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 2.4 2.6 of Rosen Introduction I When talking
Riemann Conjecture Proof and Disproof
From the SelectedWorks of James T Struck Winter March 2, 2018 Riemann Conjecture Proof and Disproof James T Struck Available at: https://works.bepress.com/james_struck/72/ Disproving and Proving the Riemann
Math 229: Introduction to Analytic Number Theory Čebyšev (and von Mangoldt and Stirling)
ath 9: Introduction to Analytic Number Theory Čebyšev (and von angoldt and Stirling) Before investigating ζ(s) and L(s, χ) as functions of a complex variable, we give another elementary approach to estimating
#A61 INTEGERS 14 (2014) SHORT EFFECTIVE INTERVALS CONTAINING PRIMES
#A6 INTEGERS 4 (24) SHORT EFFECTIVE INTERVALS CONTAINING PRIMES Habiba Kadiri Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Canada habiba.kadiri@uleth.ca Allysa
Primes go Quantum: there is entanglement in the primes
Primes go Quantum: there is entanglement in the primes Germán Sierra In collaboration with José Ignacio Latorre Madrid-Barcelona-Singapore Workshop: Entanglement Entropy in Quantum Many-Body Systems King's
EXTENDED RECIPROCAL ZETA FUNCTION AND AN ALTERNATE FORMULATION OF THE RIEMANN HYPOTHESIS. M. Aslam Chaudhry. Received May 18, 2007
Scientiae Mathematicae Japonicae Online, e-2009, 05 05 EXTENDED RECIPROCAL ZETA FUNCTION AND AN ALTERNATE FORMULATION OF THE RIEMANN HYPOTHESIS M. Aslam Chaudhry Received May 8, 2007 Abstract. We define
Pretentiousness in analytic number theory. Andrew Granville A report on joint work with K. Soundararajan, and Antal Balog
Pretentiousness in analytic number theory Andrew Granville A report on joint work with K. Soundararajan, and Antal Balog The number of primes up to x Gauss, Christmas eve 1849: As a boy of 15 or 16, I
Arithmetic Statistics Lecture 1
Arithmetic Statistics Lecture 1 Álvaro Lozano-Robledo Department of Mathematics University of Connecticut May 28 th CTNT 2018 Connecticut Summer School in Number Theory Question What is Arithmetic Statistics?
On a diophantine inequality involving prime numbers
ACTA ARITHMETICA LXI.3 (992 On a diophantine inequality involving prime numbers by D. I. Tolev (Plovdiv In 952 Piatetski-Shapiro [4] considered the following analogue of the Goldbach Waring problem. Assume